Question

$\frac{1}{1!(n-1)!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+\dots =$

• A. $2^n$
• B. $2^n/(n-1)!$
• C. $2^n/n!$
• D. $2^{n-1}/n!$

Answer 1

Answer: (D)

$S=\frac{1}{1!(n-1)!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+\cdots$
Multiplying each term by $n!/n!,S$ reduces to
$S=\frac{1}{n!}\left[\frac{n!}{1!(n-1)!}+\frac{1}{3!}\frac{n!}{(n-3)!}+\frac{1}{5!}\frac{n!}{(n-5)!}+\cdots \right]$
$=\frac{1}{n!}\left[{}^n{C}_1+{}^n{C}_3+{}^n{C}_5+\cdots \right]$
$=\frac{2^{n-1}}{n!}$